Granular Matter (2014) 16:115–124
DOI 10.1007/s10035-013-0469-x
ORIGINAL PAPER
Scalings of axisymmetric granular column collapse
J. M. Warnett · P. Denissenko · P. J. Thomas ·
E. Kiraci · M. A. Williams
Received: 4 March 2013 / Published online: 7 December 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract Experimental investigations into the collapse of
granular columns are presented with dimensional results of
the final deposit obtained using a 3D laser scanner. The high
accuracy measurement method has found that the final pile
radius is not only dependent on the aspect ratio of the initial
geometry as previously thought, but also the initial column
radius and hence the ratio of column radius to particle diameter. This was also observed to be true for the angle at the
base of the deposit. Theoretical considerations coupled with
obtained data have allowed the angle at summit for aspect
ratios less than approximately 3 to be described entirely in
terms of material parameters and aspect ratio—something
desired for all length scalings but seemingly unobtainable
due to the dynamic nature of the collapse. With the method
being non intrusive, measurement of the summital radius was
also made possible. The final height showed a dependence
predominantly with aspect ratio in agreement with previous
authors.
Keywords Granular collapse · Granular flow · Failure
surface · Avalanches
1 Introduction
Granular flows have been the interest of much research due
to their existence in many man made processes and natural
J. M. Warnett (B) · P. Denissenko · P. J. Thomas
School of Engineering, University of Warwick, Gibbet Hill Road,
Coventry CV4 7AL, UK
e-mail: j.m.warnett@warwick.ac.uk
E. Kiraci · M. A. Williams
Warwick Manufacturing Group, University of Warwick,
Gibbet Hill Road, Coventry CV4 7AL, UK
phenomena. The handling of grains and powders is important
to many civil engineering projects, such as the storage of
grains and powders, and also to the pharmaceutical industry
in the sorting of pills and medicines. Description of these
flows have been offered by interpretation of the collapse of a
granular column [1–8] and the failure of a granular step [9–
11]. The avalanching flows observed in these experiments
are analogous to some geophysical flows. Examples such as
pyroclastic flows and avalanches are of particular interest for
hazard management, but direct observation of these flows are
rare and dangerous to undertake. Predictions of such deluges
are based on the final deposits [12] and so the dynamics of
collapse can be difficult to describe over specific geographic
topologies with variability in the landscape.
The collapse of an axisymmetric granular column is an
unsteady flow; there are phases of acceleration, steady flow
and deceleration during the collapse. This dynamic is further
complicated by the existence of static and flowing regions
within the collapsing column. A general physical description of avalanching granular flows is given by Cates et al.
[13] in terms of mechanisms behind the development of a
sand pile. A sand pile can be seen as ‘fragile’ in that it cannot
elastically support some infinitesimal loads. This fragility
leads to avalanching and settles where grains have successfully jammed under the load of a granular force chain. A
slight change in the direction of the force acting on a grain
at the extremity (surface of the pile) can break this chain and
results in further reorganisation of the previously jammed
medium below and additional avalanching.
Theoretical models of granular flows have been developed
using depth-averaged and Saint–Venant equations, applied
with a varying degree of success dependent upon the flow
regime [14–17]. This is due to the lack of a set of constitutive laws as in fluidic systems. The collapse of a granular
column is a highly unsteady example of a granular flow and
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116
J. M. Warnett et al.
Fig. 1 Experimental setup. a Initial setup of granular column on rotating table. b Column after collapse with dimensions
applying these sets of equations struggled to recover the precise dynamic behaviour of the system [18]. Much progress
has been made in this area with a different combination of
models; for example Larrieu et al. [19] used a combination
of shallow water depth-averaged equations with the regular
timed addition of material to the flow. This model is in good
agreement with the scalings already found experimentally
up to an aspect ratio of a < 10, but requires an unphysical
fricition coefficient. This is in addition to the evolution of the
profile of the deposit often being exagerrated. Even greater
progress has been made in the collapses’ 2D counterpart,
the failure of a granular step. This particular collapse while
largely exhibiting similar flow regimes results in slightly different scalings, suggested to be due to the geometry dependent mechanism of side-ways mass ejection [4]. Doyle et al.
[20] adapted the model proposed by Larrieu et al. by including an estimation for the interface between the static and
flowing region which allow a more realistic coefficient of
friction. Crosta et al. [21] used a Mohr–Coulomb yield rule
with non associative flow that was in very good agreement
with the physical experiments, although struggled with the
tapering of the front of the flow. The most encouraging and
accurate model to date, again in 2D, is the μ(I ) rheology
proposed by Lagree et al. [22] recovering the evolution of
the flow with a greater accuracy. In the future it is hoped
that the μ(I ) rheology can be successfully applied to the 3D
cylindrical column collapse case.
Computationally this collapse has been investigated by
discrete-element modelling (DEM) simulations; in 2D by
Staron and Hinch [4] and Zenit [5], and in 3D by Cleary and
Frank [6]. Knowing that scalings achieved for the 2D collapse
[9–11] display variations from its axisymmetric counterpart,
comparison is best drawn from the 3D case. Cleary and Frank
demonstrate that particle shape is a limiting factor in reproducing experimental results and achieve values obtained by
both Lube et al. [1] and Lajeunesse et al. [2] for their singular
test case. Non spherical particle models, particularly in 3D,
123
consume a much larger amount of computational power and
reasons why this investigation has not been taken further.
Several previous authors [1–3] have investigated the collapse of an axisymmetric column and have concluded several simple scaling laws based on the initial aspect ratio of
the granular column. Lube et al. [1] used a vernier scale and
laser pointer technique with a stated accuracy of ±0.1 mm to
dimensionalise the final column, where as Lajeunesse et al.
[2] used a camera imaging technique with a spatial resolution of 0.4 mm. In this study a different experimental technique is applied where a 3D laser scanner with a substantially
higher accuracy of ±0.04 mm is employed for data acquisition with the added benefit of digitizing the result for analysis
and further interrogation. Experiments performed using this
technique with similar initial system sizes to those of previous authors have revealed a variation of previously proposed
dimensional scalings with a dependence on this initial system
size. The experimental method has also allowed investigation
into the angular profile of the deposit and the rate at which
the summit is consumed with an increase in aspect ratio.
2 Experimental setup
2.1 Apparatus
The apparatus was set up as in Fig. 1. A cylinder of radius
r0 was placed on a smooth perspex plane and partially filled
with granular media of mass m 0 to a specified height h 0 .
The aspect ratio, a = h 0 /r0 , was varied for a single cylinder
radius r0 by varying the mass of the particulate, m 0 . A number
of cylinder radii were used with a range of aspect ratios as
outlined in Table 1. The experiments were performed in a
randomised order to prevent systematic errors from affecting
the final results.
In these experiments the granular media used was a limestone particulate with bulk density ρ = 1.5 g/cm3 and a
Axisymmetric granular column collapse
117
Table 1 Range of aspect ratios a trialled for each cylinder of radius r0
and associated r0 /d value
r0 (mm)
r0 /d
a range
20
28
0.24–8.69
25
35
0.30–7.00
30
42
0.20–6.05
35
50
0.24–5.29
40
57
0.25–4.40
45
64
0.24–4.02
75
107
0.28–1.69
dynamic angle of repose of θr ≈ 30◦ . The angle of repose
is the maximal angle to the horizontal sustained by a pile
of granular material before the slope face begins to slide.
This value was given by the manufacturer and was confirmed
experimentally by slowly funneling the particulate onto a horizontal surface and finding the greatest slope of the resulting
deposit.
To ensure repeatable initial conditions grains were sieved
prior to filling to select grains of diameter d = 0.6 − 0.8 mm,
which were then funneled into the cylinder and the top gently flattened. This results in a non-dimensional system size
parameter defined as r0 /d. The mean packing density of the
material in the cylinder is calculated φ = m 0 /(ρπr02 h 0 ), and
found to be 0.78–0.82.
The cylinder was connected to rope over a series of pulleys
that was mounted above the table. The cylinder was removed
quickly by pulling on the rope at a speed greater than the
speed of collapse to ensure minimal interference with the
collapse itself. An average speed of 1.8 ms−1 was found to
be sufficient. After release, the material spreads across the
table resulting in a final deposit.
A 3D scanner was then used to digitize the resulting
deposit enabling precise dimensionalisation of the pile by
use of the analysis software ‘Geomagic Qualify’ (3D Digital
2002; Geomagic, Research Triangle Park, NC, USA). This
enabled non-contact measurement with high accuracy and a
greater amount of interrogation in comparison to previous
studies.
2.2 Data capture and scanner calibration
The 3D laser scanner used in this study was the ‘Nikon
Metrology MCA 2400 M7’ articulated arm enabling complete detailed geometric scans of the sand piles. The scanner
head produced a laser stripe that was made to traverse the
deposit, capturing data points viewable in real time providing immediate feedback on the scanning pace, progress and
scanned area. The scanner was able to scan areas as large
as 470 mm × 470 mm which was a limitation in terms of
allowable aspect ratio ranges shown in Table 1.
A typical scan path consisted of 1,000 measurement points
per stripe. To maintain a consistent number of measurement
points per scan path and reduce the level of data noise it is
critical to sustain the optimum distance between the scanning
head and deposit. The scanner produced a laser guide with
the stripe enabling the laser user to maintain this distance and
obtain quality data. With this level of precision, the obtained
point cloud contained 800–900 points per cm2 . When using
the equipment to scan objects with defined edges, numerous
scans can be easily amalgamated to produce the final object.
The software was found to struggle performing this task with
granular heaps due to the largely uneven surface and so all
scans were completed in a single motion of the articulated
arm.
The laser intensity also has an impact on the quality of
data achieved, and this is managed by the equipment itself to
include adjustments according to lighting conditions. Once
the system is fully calibrated the lighting levels were maintained and temperature controlled at ±5 ◦ C relative to the
temperature when calibrated. Single point and length accuracy of the scanner is 31 microns and 42 microns ±2σ respectively. These values were determined by point repeatability
and volumetric accuracy tests as outlined in the industrial
standard ASME B89.4.22 [23,24].
2.3 Processing and accuracy
The point cloud generated was interpreted as a surface mesh
as in Fig. 2 using ‘Geomagic Qualify’; industrial standard
software used for the inspection of scanned objects and materials. On occasion particles can be ejected a measurable
distance further from the edge of the resultant pile. These
were easily identifiable and removed from the final mesh for
evaluation.
The global coordinate system of the mesh was altered
so the table that the deposit rested on aligned with the x-y
plane. Four 2D cross sections were taken of the resultant mesh
through the centre of the pile. The ability to take several 2D
cross sections of the final deposit allowed evaluation of the
initial setup and cylinder removal. The cross sections made it
easy to find where the cylinder had not been raised vertically
as the resultant pile would be asymmetric, so adjustments
could be made to the equipment in trials before the main
experiments were run.
The achieved cross sections were then used to retrieve
several dimensions of the final deposit indicated in Fig. 1; the
final pile radius, r f , the final pile height, h f , the summital
angle, αs , the angle at the base, αb , and in the case of a
truncated cone, rs , the summit radius. In the case of all these
measurements with the exception of the final pile height, an
average was taken across all four 2D cross sections where
there is known to be variability due to an uneven periphery.
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118
Fig. 2 Example surface mesh for r0 = 45 mm a = 1.87 produced
from point cloud achieved with laser scanning of the resultant deposit.
Boundary effect demonstrated by two orange periphery lines formed
J. M. Warnett et al.
from the inner most and outer most points. The distance between these
boundaries is 3.30 mm, approximately 4.7d
Fig. 3 Two distinct morphologies arise dependent upon aspect ratio a. a Truncated cone when a < 0.90. b Sloping conical shape when a > 0.90
Repeatability testing was performed on several initial
aspect ratios and cylinder sizes to determine the maximal
variation in dimensional quantities. The largest source of
error within a single collapse can be observed at the periphery
of the pile where grains do not fall into a clearly defined edge.
Figure 2 highlights the smallest and largest possible boundaries in a particular example while displaying the deviation
from the possible edges. Across varying aspect ratios this
was observed to be 3d (±1.1 mm) where a < 0.50 up to 5d
(±1.8 mm) for a > 2, confirmed by inspection of the resultant scans. Averaging across the four 2D cross sections in this
way finds the mean periphery profile of the deposit. Results
from the repeatability testing and accounting for the error
due to the uneven rim of the deposit, dimensional quantities
were found to vary as r f = ±1.6 − 2.3 mm (aspect ratio
dependent), rs = ±2.0 mm, h f = ±0.5 mm, and angular
quantities ±1◦ . The material was measured to within ±0.1 g,
allowing only a variability in packing.
3 Experimental observations
3.1 Flow description and morphology
Qualitative descriptions of the collapse have been previously
studied by other authors [1,2]. For a 3 the periphery of the
column crumbles vertically downwards resulting in a frontal
123
flow to develop at the foot of the column that propagates
radially outwards. Subsequently collapsing layers flow over
the surface of the deposit continuing the flow in the radial
direction. This results in a secondary front which separates
the frontal flow and the currently static summittal region,
which propagates inwards. This results in a circular discontinuity between the two regions, which is eventually consumed by the avalanche and the flow continues until stability
is achieved in the summital region of the deposit.
In this range two distinct morphological deposits that can
be seen in Fig. 3. For shallow columns where a < 0.90 the
summit is never completely consumed by the avalanche and
results in a truncated cone. Upon exceeding this aspect ratio
the summit is entirely consumed and a full conical shape
is left. Previous authors have found this critical aspect ratio
to be a = 0.74 [1,2]. Without direct inspection with a 3D
scanner, it can be difficult to determine where one geometric
regime ends and the second begins as a summit may still
exist but is small. It is for these reasons that a higher value is
expected to be achieved, although this maybe compounded
with the difference in granular material.
With the non-intrusive accurate measurement capabilities
the rate of consumption of the summital plateau was also
observed. The variation of the summit radius, rs , with aspect
ratio is shown in Fig. 4 where rs is normalised against the
initial radius as
Axisymmetric granular column collapse
119
for some mean angular profile of the slope of the deposit θ̄ .
Some rearrangement gives
rs∗ = a tan θ̄
Fig. 4 Variation of normalised summital radius rs∗ = (r0 − rs )/r0
against aspect ratio. Dashed line indicates linear fit given in Eq. (2)
rs∗ =
r0 − rs
r0
(1)
This shows a common linear relation across all cylinder sizes
given by
rs∗ = a − 0.055
(2)
valid for a < 0.90. Where a > 0.90 there is a sharp deceleration in the shrinking of the plateau with increasing aspect
ratio as the peak becomes curved and the summital plateau
becomes indistinguishable as it is of the order of a few grain
sizes.
The direct proportionality observed is expected with consideration of geometric arguments. Under the first regime,
the collapse results in a truncated cone. Hence
rs = r0 − h 0 tan θ̄
(3)
(a)
(4)
in agreement with the obtained data.
Where a 3 the collapse dynamic changes from that
previously observed. In this case the whole upper surface of
the column moves, initially retaining its horizontal profile. A
frontal flow then develops at the base on the pile as before
spreading radially outwards. Part way through the collapse
the upper surface begins to dome while the frontal flow continues to spread. The end result is a deposit with a shallower
angular profile than lower aspect ratios.
The evolution of the profile with increasing aspect ratio
can be seen in Fig. 5a for r0 = 35 mm. The height, h(r )
is given as a function of radius, r , and both values are normalised against r0 . It is evident that the angular profile not
only becomes shallower with increasing a, but the nose of the
deposit becomes less sharp and displays an increase in curvature. The relationship between defining dimensional values
and aspect ratio are discussed in detail below to include the
final deposit radius, r f , final height, h f , and angles at the
summit, αs , and base, αb . This has previously been investigated by Lube et al. [1] and Lajeunesse et al. [2] but a further
dependence has emerged; a dependence on the initial system
size r0 /d.
Similar system sizes were used as in previous experimental works, but notable differences in the defining dimensional
values were found and are presented in this study. An example
of resultant deposit profiles for a = 1.7 is given in Fig. 5b for
several cylinder sizes. In this particular case the final height
and angle at the summit is largely the same. The difference
occurs at the nose of the deposit where larger system sizes
result in a greater final deposit radius and a shallower angle
(b)
Fig. 5 Deposit profiles normalised against the cylinder radius r0 . a Evolution of the profile with increasing aspect ratio a for r0 = 35 mm.
b Difference in profiles for a = 1.70
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120
J. M. Warnett et al.
(a)
(b)
Fig. 6 Relationship between r∗ and a for varying cylinder radii r0 with coefficients given in Table 2. a a < 1.7 with errors displayed as discussed
in Sect. 2.3. Linear fits for each r0 satisfying Eq. (6) are shown. b a 1.7 following a power law relation as indicated by line fit
at the base. The impact of r0 /d is considered for each of
the dimensional quantities and frequently displays an arresting value of this effect. Meso-scale interactions are known
to occur across granular systems but in this case it has been
previously overlooked.
3.2 Final pile radius
Previous results [1–3] suggested that the final deposit radius
is independent of the initial system size. Final pile radius, r f ,
against aspect ratio was considered for each cylinder radius,
with r f normalised against r0 as
r∗ =
r f − r0
r0
(5)
First consider where a < 1.7 as shown in Fig. 6. A linear
relation of the form
(6)
where P1 and Q are constants is observed, but there is a
clear dependence of these coefficients on the initial cylinder
size and inherently the system size. The gradient P1 for each
cylinder radii is given in Table 2. It evidently increases with
cylinder radii although it indicates strong signs of arresting.
For r0 = 20 − 45 mm it follows a strongly linear relation
given by:
P1 = 0.013
123
r0
+ 0.45
d
r0 (mm)
r0 /d
P1
P2
20
28
0.79
1.00
25
35
0.88
1.21
30
42
1.02
1.25
35
50
1.06
1.33
40
57
1.18
1.34
45
64
1.26
1.34
75
107
1.29
–
and substituting this into Eq. 6 gives
h0
+ 0.45a + Q
(8)
d
This reveals a secondary dependence on initial height h 0
while the system size r0 /d 70. The increase in P1 severly slows for the largest cylinder size and while limitations of the equipment prevent investigation of larger values
of r0 /d, it is presupposed that this deceleration continues
in strength with increases in P1 being negligible. This disputes previous research by other authors [1–3] who conclude
there is no such effect while investigating comparable values of r0 /d, although it is worth noting that the gradient
for r0 = 75 mm, P1 = 1.29, is close to the value of 1.24
achieved by Lube et al. With previous experiments subject
to greater inaccuracies in their measurement methods, differences in achieved data have previously been assumed to
be due to experimental error. The high spatial resolution in
the presented results and strong variation in P1 indicates the
existence of this dependence to be correct. Reasoning for this
r∗ = 0.013
3.2.1 Low aspect ratios
r∗ = P1 a + Q
Table 2 Variation in proportional constants P1 and P2 for relationship
between r∗ and a given by Eqs. (6) and (9) and respectively
(7)
Axisymmetric granular column collapse
121
difference in scaling can be given insight by consideration of
the degree of mass flow and process of jamming and flowing
layers as described by Cates et al. [13].
Consider a fixed aspect ratio a = h 0 /r0 for two different initial column radii r0 with r0 /d < 70. The larger r0
will initially contain more granular matter and so at the start
of the avalanche, the larger r0 /d will have a greater initial
deposit on the surface. The larger initial deposit has a greater
amount of jammed grains under a larger amount of pressure
and hence a longer granular force chain length. In both cases
the subsequently avalanching layers will disturb the force
chain but a smaller proportion of the force chain will be disturbed in the case of the larger deposit due to the stronger
jamming effect and the grains being under a greater amount
of pressure. This results in a smaller proportion of the flowing layer being entrained in the disturbed surface and hence
a smaller proportional energy loss in the flowing layer. This
allows for greater movement of the avalanching layer and
overall a greater runout. Where we observe a saturation of
the meso-scale effect and considering two different values of
r0 /d, the proportion of the force chain that will be disturbed
in the collapse will be small and comparable to other larger
values of r0 /d. For this reason it is reasonable to expect to
observe the arresting effect seen in the proportional constant
P1 of the final pile radius relation.
3.2.2 Larger aspect ratios
When a 1.7 a new relation begins to emerge with a good
fit to a the power law
r∗ = P2 a
0.66
(9)
where P2 varies for r0 /d as given in Table 2. Compared to
the relation given in Eq. (6) this constant appears to arrest
earlier having stagnated when r0 = 35 mm, r0 /d = 50. The
difference in P2 can be attributed to the earlier discussion
on jamming with its quicker cessation likely to be due to
the greater amount of material involved in the collapse. The
data is in agreement with Lube et al. in terms of the onset
of Eq. (9), but the power 0.66 is higher than both Lube et al.
and Lajeunesse et al. who agree with a square root fit. The
variation in the precise value of the power could be due to
different granular materials used to the other authors, but
again they negate the effect of r0 /d as the data clearly shows.
3.3 Final height
The final height was normalised as h ∗ = h f /r0 and compared against a in Fig. 7. In the first geometric regime where
a < 0.90 the deposit is a truncated cone and so h f = h 0 .
This suggests that there is an internal conical surface over
which the avalanche occurs, with the angle at the base of this
internal cone to be θ = tan−1 (0.90) = 42.0◦ . This obser-
Fig. 7 Evolution of normalised height h ∗ with aspect ratio a. Dashed
line indicates power law relation given by Eq. (10)
vation was concluded by Lajeunesse et al. [2] where layers
of coloured sand were used in the prepared column and the
resultant deposit was split to reveal a conical zone where
there had been no movement of material.
When entering the second geometric regime, the particulate above a height 0.90r0 avalanches over the currently deposited matter and the height minutely continues
to increase. It argued that this increase is marginally smaller
for r0 = 20, 25 mm with exponents of 0.05 and 0.07 respectively, while for all other cylinder radii the relation
h ∗ = 0.90a 0.09
(10)
holds. In this case the meso-scale effect is observed for
r0 /d 40, likely to be arresting faster than the radial coefficient P1 due to the greater amount of material as presupposed for the radial coefficient P2 . Lajeunesse et al. suggest
that there is no increase in height while Lube et al. agree that
there is, although proposing a higher exponent of 0.17.
Mohr-Coulomb theory states that the development of surface failure is due to the inability to sustain its composition
under shear stresses and occurs along an envelope projected
at an angle equal to the internal friction angle θμ . The data
provided for the limestone based particulate used indicated
an internal friction angle θμ ≈ 39◦ . This would imply that
the onset of the second geometic regime would occur at an
aspect ratio of a = 0.89 which is in good agreement with
the experimental data. This allows description of the height
with known material parameters:
h∗ = a
h ∗ = tan (θμ )a
a < tan θμ
0.09
(11)
a > tan θμ
where 0.09 is exchanged for a lower value for the smallest r0 /d. Obtaining scalings where constants are completely
123
122
J. M. Warnett et al.
(a)
(b)
Fig. 8 Variation in angle against aspect ratio for a summital angle, b base angle
described by the material parameters is desirable for equivalence in other experimental setups with different granular
materials. This is an encouraging step to a fuller description of the final deposit independent of seemingly arbitrary
proportional constants.
3.4 Angular profiling
Using the angular analysis of the onset of the second geometric regime and assuming the summital angle to be a function of aspect ratio, αs (a), we have the condition
αs (tan θμ ) = θr
(13)
Applying this condition to Eq. (12) the coefficient can be
expressed in terms of the internal parameters:
tan θμ 0.33
(14)
αs = θr
a
The surface of the deposit does not have a straight edge as a
cone does, but it curved to some angular profile. Two characteristic angles of the profile are the angle at the summit,
αs , and the angle at the base, αb . These values were obtained
from the 3D meshes generated, so the method was not intrusive or destructive of the deposit. Where the first geometric
regime exists and there is a flat summit, αs was taken to be
the angle of the slope directly below the flat surface.
For a 3 the summital angle was found to vary with
aspect ratio as in Fig. 8a, following a power relation:
Enabled by the previous analysis via interpretation of Mohr–
Coulomb theory, this is the second scaling with constants
given in terms of known material parameters.
When a 3 a different relationship emerges coinciding
with the change in collapse regime where the entire upper
surface of the initial column falls retaining its horizontal profile. The angle at the summit continues to decrease but at a
severely slower rate following the power law
αs = 25.6a −0.33
αs = 20.1a −0.08
(12)
While the aspect ratio is such that the resulting deposit is a
truncated cone, the avalanching periphery is strongly dependent upon the internal friction angle rather than the angle of
repose as discussed in Sect. 3.3 and so αs > θr . The onset
of the second geometric regime where the entirety of the
summit is consumed corresponds to the aspect ratio where
material exists above the conical failure envelope defined by
the internal friction angle. At aspect ratios greater than this
critical onset, the greatest sustainable angle of a deposit with
a sharp peak is exactly the angle of repose by definition.
Hence where this occurs αs < θr . The continued decrease
in the summital angle is due to increased avalanching over
already deposited layers smoothing the steep sides.
123
(15)
The variation in the angle at the base with aspect ratio was
found to be more complex and dependent upon the r0 and
hence the initial system size as shown in Fig. 8b. While a 3
this follows a power law relation of the form
αb = Ea −F
(16)
where E and F are constants dependent on r0 /d as given
in Table 3. The dependence on r0 /d is initially linear up to
r0 = 45 mm, giving relations
r0
E = 14.7 − 0.066
(17)
d
r0
(18)
F = 0.010 − 0.78
d
Axisymmetric granular column collapse
123
Table 3 Constants fitting power law relation αb = Ea −F for given
initial cylinder size
r0 (mm)
r0 /d
E
F
20
28
12.94
0.52
25
35
12.33
0.45
30
42
11.69
0.32
35
50
11.42
0.25
40
57
10.98
0.23
45
64
10.53
0.17
75
107
9.76
0.16
For r0 = 75 mm the F exponent evidently stagnates and it
can be assumed that for r0 /d > 100 F = 0.16. With an error
of ±1◦ it is presupposed that E ≈ 10 for r0 /d > 100. For
a 3, αb stagnates with 6◦ < αb < 8◦ .
For a fixed aspect ratio, smaller system sizes have a distinctly higher base angle at least in the first geometric regime.
This would be expected given a proportionally smaller radius
of the deposit and equivalently low r0 /d as described in
Sect. 3.2. After the end of the first geometric regime the
variation in angle between cylinder sizes becomes small and
approaches the stagnation point. Within this range, avalanching of the upper surface of the column flows over the already
deposited layers that stretch beyond 0.75 cylinder radii and
are of similar thickness at the extremity. αb < θr for all aspect
ratios as expected or further avalanching would occur as per
the definition of the angle of repose.
4 Conclusions
The methodology used in the experiments have allowed a
substantially greater degree of accuracy than any previous
research on the subject. Coupled with mesh generation of
the deposits and the ability to obtain data non-intrusively, a
deeper dependence on the scaling of granular collapse has
been revealed. It was previously thought that in the absence
of internal parameters, scalings of the collapse were uniquely
describable in terms of the aspect ratio of the initial granular column. Given similar initial conditions to the previous
studies, this analysis has shown that differences in data that
were originally explained as experimental error are a demonstration of system size dependence calculated as r0 /d. While
this dependence exists, it has demonstrated a saturation at
the larger r0 /d with this particular point dependent on the
scaling under consideration. Furthermore some relations are
expressible in terms of internal parameters, desirable of all
scalings for a full theoretical model of the collapse to exist.
It is well known that the final deposit radius has a linear dependence on aspect ratio, but results showed that the
coefficient of this proportionality increased, at least initially,
with the system size. For values r0 /d 70 this proportionality was also linear, showing signs of abruptly saturating for
r0 /d = 107. This behavior is mirrored in the angle at the
base of the deposit; larger initial system sizes have a shallower base angle. Together these results support the idea that
the initially deposited layers have an effect on the scalings of
the resultant pile, with their deposition dependent upon the
amount of initial mass collapse and the resultant jamming.
For a fixed aspect ratio, a greater initial radius means a thicker
initial deposit. The thicker deposit experiences greater jamming, and hence results in greater spreading of subsequently
avalanching material and a shallower angle at the base of the
resultant pile.
In the first geometric regime a summital plateau exists
that shrinks with increasing aspect ratio. The continued contraction was directly proportional to aspect ratio, but had no
relation to system size or rate of increase of the final pile
radius. Within this first geometric regime the final height is
exactly the initial height, but where a > tan θμ the increase in
height significantly slows of the order a 0.09 . It is argued that
the height increases even slower in the smaller systems with
r0 /d = 28, 35. The angle at the summit follows a power law
relation with aspect ratio, independent of system size, and
coupled with other analysis is shown to depend on the angle
of repose and internal friction while a 1.7.
In summary, dimensions r f and αb that result directly from
material deposition depends strongly upon the initial system
size r0 /d in addition to the aspect ratio and material parameters. Dimensions rs , h f and αs which result from material
loss to avalanching and not the history of the collapse have
little or no secondary dependence and are subject only to
aspect ratio and material parameters. This further fuels the
necessity for any theoretical model of the collapse to incorporate internal dynamics such as those described by Cates
et al. [13], particularly where investigations could contain a
meso-scale effect.
Acknowledgments This research was funded by an EPSRC doctoraltraining grant to whom we offer our gratitude.
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